Mixed Models for Finite Populations Mixed models are widely used to develop inferences that lead to disease prevention in clustered and longitudinal studies of public health. Typical mixed model methods do not account for the finite size of clusters or the finite number of clusters in populations. Recently, predictors of realized random effects based on finite population mixed models have been developed, and shown to have smaller expected mean squared error than the usual predictors. The new methods are innovative since they provide a non-parametric design based framework for optimal prediction. They are broadly applicable to biomedical research, e.g., cluster randomized trials and clustered or longitudinal sample surveys! The results, however, are currently only developed for equal cluster size populations where equal size samples are selected from selected clusters. This research will develop and extend design-based finite population mixed models, evaluate estimators and predictors obtained under such models relative to competitors, and apply such methods to cluster randomized trials and longitudinal studies. More specifically, the research will: a) extend the finite population mixed model to include replicated response error, missing data, and response error in populations with unequal cluster sizes. b) Develop finite population mixed model methods to include cluster/unit level auxiliary variables to improve predictors of realized random effects, and to develop estimators of regression parameters for auxiliary variables. c) Develop and extend finite population mixed model methods to longitudinal settings, and settings where units are observed under more than one condition. This will be accomplished by developing new predictors (estimators) for samples from finite populations, comparing properties of the predictors relative to competitors from the literature, developing empirical predictors along with software for their implementation, and evaluating the performance of empirical predictors via simulation studies. [unreadable] [unreadable]